Hemicompact space

In mathematics, in the field of topology, a topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set in the sequence.^[1] Clearly, this forces the union of the sequence to be the whole space, because every point is compact and hence must lie in one of the compact sets.

Examples

Every compact space is hemicompact.
The real line is hemicompact.
Every locally compact Lindelöf space is hemicompact.

Properties

Every hemicompact space is σ-compact and if in addition it is first countable then it is locally compact. If a hemicompact space is weakly locally compact, then it is exhaustible by compact sets.

Applications

If $X$ is a hemicompact space, then the space $C(X,M)$ of all continuous functions $f:X\to M$ to a metric space $(M,\delta )$ with the compact-open topology is metrizable.^[2] To see this, take a sequence $K_{1},K_{2},\dots$ of compact subsets of $X$ such that every compact subset of $X$ lies inside some compact set in this sequence (the existence of such a sequence follows from the hemicompactness of $X$ ). Define pseudometrics

d_{n}(f,g)=\sup _{x\in K_{n}}\delta {\bigl (}f(x),g(x){\bigr )},\quad f,g\in C(X,M),n\in \mathbb {N} .

Then

d(f,g)=\sum _{n=1}^{\infty }{\frac {1}{2^{n}}}\cdot {\frac {d_{n}(f,g)}{1+d_{n}(f,g)}}

defines a metric on $C(X,M)$ which induces the compact-open topology.

Notes

^ Willard 2004, Problem set in section 17.
^ Conway 1990, Example IV.2.2.

References

Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
Conway, J. B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96. Springer Verlag. ISBN 0-387-97245-5.

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[1] Willard 2004, Problem set in section 17.

[2] Conway 1990, Example IV.2.2.